proposition 44.11 The boundary term, and the Gibbons–Hawking–York action

Rests on

• depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
  • depends_on proposition 5.2 Jacobi's formula, column form ¶
    • depends_on equation 5.19 eq:lin-leibniz-det ¶
    • depends_on proposition 7.30 Leibniz rule ¶
      • depends_on definition 7.12 Product of functions ¶
        • depends_on definition 7.9 Real function ¶
      • depends_on equation 7.14 eq:ana-derivh ¶
      • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-11 ¶
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • proves proof ch:05-real-analysis@proof-14 ¶
    • depends_on proposition 5.49 Injective, surjective, invertible ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
      • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
        • depends_on definition 5.47 Inverse of a linear transformation ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-17 ¶
      • depends_on theorem 5.40 Rank–nullity ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on lemma 5.38 Exchange and completion ¶
        • proves proof ch:03-linear-algebra-representations@proof-13 ¶
      • proves proof ch:03-linear-algebra-representations@proof-18 ¶
    • proves proof ch:03-linear-algebra-representations@proof-1 ¶
  • depends_on theorem 7.133 Gauss ¶
    • depends_on definition 7.127 Simple regions ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.20 Continuity at a point ¶ ↺
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • depends_on definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
    • depends_on remark 7.128 What the derivations below take as given ¶
      • depends_on definition 7.125 Multiple integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶
      • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
        • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • proves proof ch:05-real-analysis@proof-23 ¶
      • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
    • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      • depends_on definition 7.41 Antiderivative ¶
        • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
        • depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
        • depends_on theorem 7.24 Extreme value theorem ¶ ↺
        • proves proof ch:05-real-analysis@proof-24 ¶
      • proves proof ch:05-real-analysis@proof-25 ¶
    • proves proof ch:05-real-analysis@proof-79 ¶
  • proves proof ch:03-einstein-field-equations@proof-5 ¶
• depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
  • depends_on definition 44.6 Stress–energy tensor ¶
    • depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      • depends_on equation 16.31 eq:calcvar-field-functional ¶
      • depends_on lemma 16.18 Fundamental lemma ¶
        • depends_on definition 16.15 Variation; the first variation ¶
        • depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
        • proves proof ch:14-calculus-of-variations@proof-8 ¶
      • depends_on theorem 7.133 Gauss ¶ ↺
      • proves proof ch:14-calculus-of-variations@proof-20 ¶
  • depends_on lemma 44.7 Variation of inverse and determinant ¶
    • depends_on proposition 5.2 Jacobi's formula, column form ¶ ↺
    • proves proof ch:03-einstein-field-equations@proof-2 ¶
  • depends_on lemma 44.8 Palatini identity ¶
    • depends_on definition 13.149 Metric compatibility ¶
      • depends_on definition 13.145 Affine connection ¶
        • depends_on definition 13.90 Mixed tensor ¶
        • depends_on definition 13.82 Vector field ¶
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
        • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.84 Tensor ¶
      • depends_on definition 13.146 Parallel transport and autoparallels ¶
        • depends_on definition 13.145 Affine connection ¶ ↺
    • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      • depends_on definition 13.145 Affine connection ¶ ↺
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      • depends_on definition 13.149 Metric compatibility ¶ ↺
      • depends_on definition 13.147 Torsion ¶
        • depends_on definition 13.145 Affine connection ¶ ↺
      • proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
    • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      • depends_on definition 13.145 Affine connection ¶ ↺
      • depends_on definition 13.147 Torsion ¶ ↺
      • depends_on proposition 7.105 Clairaut–Schwarz ¶
        • depends_on definition 7.20 Continuity at a point ¶ ↺
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-64 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
    • proves proof ch:03-einstein-field-equations@proof-3 ¶
  • proves proof ch:03-einstein-field-equations@proof-4 ¶
• proves proof ch:03-einstein-field-equations@proof-6 ¶

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